Limit for Markov Chains

Show that [itex]P^n[/itex] has no limit, but that: [itex]A_n=\frac{1}{n+1}(I+P+P^2+\ldots+P^n)[/itex] has a limit.

3. The attempt at a solution

I can see that [itex]P^{EVEN}=\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right)[/itex] and [itex]P^{ODD}=\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)[/itex], so a steady state is never reached, but I can't figure out the second part.